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The escape from conjectural variations: the consistency condition in duopoly theory from Bowley to Fellner

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The paper covers the debate over the conjectural variations approach to duopoly in the period going from 1924 to 1949 and focuses on the evolution of the economists’ views about the imposition of a consistency condition upon the firms’ conjectures.
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  The escape from conjectural variations:the consistency condition in duopolytheory from Bowley to Fellner Nicola Giocoli* The paper covers the 1924–1949 debate on the conjectural variationsapproach to duopoly theory and focuses on the evolution of economists’ viewsabout the imposition of a consistency condition on the firms’ conjectures. Themain point is that, although the consistency condition entailed a notion of interactive equilibrium that resembled the modern correct conjectures equi-librium, most neoclassical economists of the time refused to apply it because of the excessive requirements it imposed upon the firms’ forecasting abilities,as well as its failure to encompass an explanation of the process leadingto equilibrium.  Key words : Duopoly, Conjectural variations, Cournot model, Stackelbergmodel, Consistent conjectures.  JEL classifications : B21, D43 1. Introduction This paper covers the debate on the conjectural variations approach to duopolisticcompetition in the period from 1924 to 1949. The focus is on the evolution of economists’ views about the imposition of a consistency condition upon firms’conjectures in order to obtain a determinate solution to the duopoly model. The pointthat I wish to establish is that, although the consistency condition brought to duopolytheory a notion of interactive equilibrium that—looked at in retrospect—closelyresembled the modern game-theoretic idea of a correct conjectures equilibrium,central in the contemporary image of economics as a system of relations, mostneoclassical economists working in the period under scrutiny refused to apply sucha notion because of the excessive requirements it imposed upon firms’ forecastingabilities, as well as its failure to encompass an explanation of the process leading toequilibrium itself. Quite the contrary, these economists preferred to avoid any kind of  Manuscript received 25 November 2002; final version received 5 April 2004.  Address for correspondence : Department of Economics, University of Pisa, Via Curtatone e Montanara15, 56126 Pisa, Italy; email giocoli@mail.jus.unipi.it.*UniversityofPisa.Whilebearingfullresponsibilityforanyremainingmistake,IwishtothankMarcoDardi, Matthias Klaes, Hansjo¨rg Klausinger, Maria Cristina Marcuzzo, Salvatore Rizzello, AnnalisaRosselli, Neri Salvadori and two anonymous referees for their useful comments and suggestions. Thefinancial support of MIUR PRIN 2002 ‘Mathematics in the history of economics’ is gratefullyacknowledged. Cambridge Journal of Economics 2005, 29, 1–18doi:10.1093/cje/bei007 CambridgeJournalofEconomics ,Vol.29,No.1, Ó CambridgePoliticalEconomySociety2005;allrightsreserved NOT FORPUBLIC RELEASE  equilibrium theorising about the firms’ conjectures and turned instead to a case-by-case, almost a theoretical approach to oligopolistic competition.Starting from Arthur Bowley’s 1924 Mathematical Groundwork , 1 the idea beganto spread that the duopolists’ Cournot-style reaction functions had to be givena conjectural interpretation. One of the issues was that of reconciling the reactionfunctions—an allegedly dynamic concept—with the static set-up of the standardCournot model. 2 The latter ambiguously mixed a static formalization of what todaywould be called a one-shot simultaneous game with a dynamic story in terms of firms’actions, reactions and counter-reactions that would have better been modelled asa sequential game. In particular, each of Cournot’s reaction functions was formulatedas a static equilibrium notion but featured as an argument the actual output producedbytheotherfirm,akindofinformationthateachfirmcouldobtainonlyinasequentialgame. However, the feature of the Cournot model that raised most objections was yetanother one, namely, the assumption that each firm behaved as if its rival would notreact to its own move. This assumption was really puzzling because in Cournot’spseudo-dynamic story, each firm would inevitably realise that its rival was not passiveat all but did react to its moves.Bowley offered a chance to overcome both problems with his new notion of conjectural variations which generalised the Cournot assumption and, at the sametime, emphasised the conjectural nature of the reaction functions (see Section 2).Bowley’sapproachgainedfurther strengthafewyearslaterwhenanotherkeymodelof oligopolistic competition—the Stackelberg model, srcinally formulated in terms of the tangible reactions to firms’ actual output choices—underwent a similar process of ‘conjecturisation’ (see Section 3).Having dominated oligopoly theory for almost a decade, the conjectural approachcameunderattackinthemid-1930swhenseveralobjectionswereraised.Tostartwith,where did Bowley’s conjectural variations come from? How did a firm form them? If a firm was able to perform a sort of instantaneous mental experiment in order toformulate a definite expectation as to its rival’s choice, why should it not be assumedthat it was also able to anticipate the ultimate consequences of the experiment andthus to realise that the most profitable action for both duopolists was to produce thejoint monopoly output? And, if this was the case, was it not reasonable for a firm totrust the reasoning ability of its rival and conclude that it too would reach sucha conclusion? Finally, if, as in the Stackelberg model, we stuck to a dynamic set-up,how were firms supposed to revise their conjectures once they were falsified by theirrival’s reaction? Did this learning process converge to the Cournot equilibrium or tothe joint monopoly outcome?It was in order to answer at least some of these questions that leading economistslike Roy Harrod and Wassily Leontief suggested adding to Bowley’s and Stackelberg’smodels the shortcut of imposing a consistency condition upon the firms’ conjectures,namely, that each conjectural variation had to coincide with the actual reaction of therival (see Section 4). Yet the shortcut was far from conclusive. First, it called for very 1 The2ndeditionofA.C.Pigou’s Economicsof Welfare shouldalsobe mentionedinthisregard:Pigou(1924, pp. 237–8). 2 If not specified otherwise, by the term ‘Cournot model’ I mean the traditional, textbookinterpretation of A. A. Cournot’s 1838 analysis of duopoly, without however implying that this reallycaptures his thought. See Giocoli (2003B) for more details. 2 N. Giocoli  high demands upon the firms’ intellectual and forecasting ability, something thatcould not be easily conceded by most 1930s economists. 1 Second, and moreimportant, it extended to oligopoly theory a new image of economics as a disciplinethat looked only at the existence and static properties of equilibrium conditions, whileneglectingthetraditional issueofhowandwhytheequilibriumwasreached inthefirstplace. In the specific case of duopolistic competition, this entailed that the impositionof the consistency condition begged the crucial question of how firms actually formedtheir correct conjectures.It was on these grounds that other economists of diverse inclination, like RichardKahn, George Stigler and William Fellner, contested the consistency condition andwithitthewholeconjecturalvariationsapproach(seeSection5).Thecriticsfocusedonthe fundamental uncertainty that affected the duopolists’ conjectures and argued thatonlytheinvestigationoftheactualfunctioningofoligopolisticmarketscouldrevealhowin practice firms circumvented such uncertainty. These criticisms helped to pave thewayforthenewapproach,baseduponagreaterrecoursetofieldworkandtherefusaltosubordinate the empirical analysis to any universally valid theoretical scheme, thateventually came to dominate post-WWII industrial economics, especially in the US(see Section 6). A historically interesting implication of this story is that the researchfield that represented the most obvious outlet for the application of the newlyformulated game-theoretic notions (above all, the 1950 Nash equilibrium), remainedfor a few more decades a rather hostile environment for the more mathematically-orientedkindofeconomicsthatemergedfromtheso-calledformalistrevolution. 2 This,ifanything,mayprovideapartialexplanationofwhynon-cooperativegametheorywaslargely neglected by US mainstream economists for the first 25 years of its life. 2. Bowley’s new idea In his Mathematical Groundwork , Bowley argued that in order to solve the first-orderconditions of a standard duopoly problem with quantity-competition we should need to know [ q 2 ] as a function of [ q 1 ], and this depends on what each producerthinkstheotherislikelytodo.Thereisthenlikelytobeanoscillationintheneighborhoodof thepricegivenbytheequation marginal price ¼ selling price ,unlesstheycombineandarrangewhat each shall produce so as to maximise their combined profit. (Bowley, 1924, p. 38) To grasp Bowley’s idea, consider two firms that produce a homogeneous productwith output levels q 1 and q 2 , and an aggregate output of  Q ¼ q 1 þ q 2 . 3 Provided theinvertibility conditions are met, the market price associated with this output may beexpressed in terms of the inverse demand function: p ( Q ) [  p ( q 1 þ q 2 ). Each firm i  issupposedtohaveacostfunction c i  ( q i  ), i  ¼ 1,2.Assumingthatthestrategicvariableforboth firms is the output level, firm 1’s maximisation problem is: max q 1 p 1 ð q 1 ; q 2 Þ ¼  p ð q 1 þ q 2 Þ q 1 ÿ c 1 ð q 1 Þ : ð 1 Þ 1 See for example the exchange between Hayek (1939 [1935], 1937); Morgenstern (1976 [1935])and Hutchison (1938, ch. 4), on the legitimacy of the perfect foresight assumption. On this debate, seeGiocoli (2003A, ch. 3). 2 See Blaug (1999). 3 I have modified the notation in order to make it uniform throughout the paper. See howeverfn. 17[?]. The escape from conjectural variations 3  This shows that firm 1’s profit depends on the output choice of firm 2. In order tomake an informed decision, firm 1 must therefore forecast firm 2’s choice. A similarproblem can be formulated for firm 2.AccordingtothestandardversionoftheCournotassumption, eachfirmexpectstheother not to modify its behaviour as the market price changes. The first orderconditions (FOCs) of the firms’ maximization problems are: ¶ p i  ð q 1 ; q 2 Þ ¶ q i  ¼  p ð Q Þ ¼  p # ð Q Þ q i  ÿ c # i  ð q i  Þ ¼ 0 ; i  ¼ 1 ; 2 : ð 2 Þ These two FOCs characterise what I shall call the basic Cournot duopoly model.Generally speaking, modern microeconomics requires that in order for the firms’choices to constitute an equilibrium, two conditions need be satisfied. The firstcondition is that no firm, on the basis of its own beliefs, must desire to modify itschoice. Thesecondconditionisthattheequilibriumactionsofthefirmsareconsistentwith the beliefs upon which they act. Thus, in the duopoly model an equilibrium isgivenbyevery pair of output levels ð ^ q 1 ; ^ q 2 Þ such that: (i) each firm is choosing its profitmaximising output given the beliefs about the other firm’s choice; and (ii) each firm’sbeliefs are correct at equilibrium. In our model such an equilibrium pair identifies theCournot equilibrium.Although it is now customary to characterise the duopolists’ reaction functions(RFs)intermsofeachfirm’sbeliefsaboutitsrival’schoice,beforeBowley’sinnovationthe standard interpretation saw the interaction between the duopolists as taking placesequentially, so that each RF was a relation determining a firm’s action in a givenperiod in terms of the other firm’s action during the preceding period. According tothis view, the reaction function of firm 1 depicts how firm 1 will modify its outputchoice according to its rival’s choice q 2 . Given that each FOC determines the optimalchoice as a function of the rival’s output, firm 1’s RF f  1 ( q 2 ) is implicitly defined by theFOC ¶ p 1 ð  f  1 ð q 2 Þ ; q 2 Þ ¶ q 1 ¼ 0 : A similar equation holds for firm 2’s RF f  2 ( q 1 ). The RFs werethus static equilibrium notions applied to a sequential set-up. Bowley’s new idea wasprecisely to solve the tension between the statics and the dynamics of duopolisticcompetition by giving a conjectural interpretation of the reactions functions. Thisentailed the possibility (although not the necessity) of considering the duopolists’interaction as a one-shot, simultaneous choice set-up. 1 The conjectural variation (CV) v ij  ¼ ¶ q  j  ¶ q i  represents firm i  ’s conjecture about how j  will respond to a small variation of  i  ’s output. Call v 12 the arbitrary conjecture thatfirm 1 formulates over firm 2’s conduct. Firm 1’s FOC becomes: ¶ p 1 ð q 1 ; q 2 Þ ¶ q 1 ¼  p ð Q Þþ  p # ð Q Þ½ 1 þ v 12  q 1 ÿ c # 1 ð q 1 Þ ¼ 0 : ð 3 Þ The same reasoning can be repeated for firm 2, whose conjectural parameter is v 21 .Bowley’s introduction of the term ¶ q  j  / ¶ q i  in the FOCs was a major novelty withrespect to the usual representation of the duopoly problem. This term meant that themodel’s solution depended ontheexactvalueofeachfirm’sconjecture about itsrival’sreaction. Hence, if  v 12 ¼ v 21 ¼ 0 we obtain the FOCs of the Cournot model, whereeach firm believes that its rival will not react to one’s own choice; if  v 12 ¼ v 21 ¼ ÿ 1 we 1 See Friedman (1977, p. 149). 4 N. Giocoli  havethecompetitivemodel,wheretheFOCisnothingbutthestandardmarginalcost-pricing rule; if  v ij  ¼ q  j  / q i  , i  , j  ¼ 1, 2, we get the joint monopolyoutcome (with in case of identical firms).That the conjectural variations approach could capture in a single parameter theintensity of firms’ rivalry and encompass as special cases the classic duopoly modelsentailed,ontheonehand,that,intheabsenceofany a priori  constraintuponthefirms’conjectures, the duopoly equilibrium was actually undetermined. On the other hand,this feature gave economists the freedom to devise values for the CV terms thatwarranted the desired solution. In short, Bowley’s proposal showed that the ghost of indeterminacy still hung over the duopoly model, but only up to a proper assignmentof values to the CV terms. 1 Thus, the CVs seemed to provide the Holy Grail of oligopoly theory, namely, that unitary approach to the topic that had been aspired toby more than one generation of economists. It is noteworthy that in such an approach,firms’ conjectures were explicitly under the spotlight.Unfortunately, we now know that the CVs cannot constitute a satisfactory methodfor tackling firms’ behaviour. The reason is that they fall short of eliminating theconfusion between statics and dynamics. Each CV term, in fact, indicates that one of thefirmsexpects theother toreactinsomespecific waytoitsownchoice. Buthowcanthe rival react if the interaction takes place just once and simultaneously? Either wehave an explicitly sequential set-up or we have to admit that the only reasonable CV isthe Cournot one: the rival is expected not to react simply because the game ends withthe simultaneous moves. Moreover, the CVs are quasi-dynamic concepts whosemeaning and use in a dynamic setting together with static equilibrium notions like theRFs is highly questionable. 2 It is historically remarkable that these objections to the CV approach were raisedquite soon, as early as the 1930s. Yet, as Cournot himself had anticipated (Cournot,1971 [1838], p. 83), the most immediate effect of letting the mental variables in wasthat of enhancing the plausibility of the joint monopoly solution. In a review of Bowley’s Mathematical Groundwork , Allyn Young argued that if we allowed theconjectural element to enter the analysis, we also had to concede that each firm couldanticipate the ultimate consequences of its rival’s chain of adjustments and thusdiscover that they were less profitable than the joint monopoly outcome. Collusionwould then turn out to be the stable solution of the duopoly model because eachperfectly rational duopolist would understand that deviating from it would causelosses to both firms (Young, 1925, p. 134). Similarly, in a 1928 paper JosephSchumpeter argued that intelligent duopolists could not fail to realise all theimplications of their situation, so that ‘they will hit upon, and adhere to, the pricewhich maximises monopoly revenue for both taken together. [ . . . ] The case will notdiffer from the case of conscious combination—in principle—and be just asdeterminate’ (Schumpeter, 1928, p. 370). As I show in the next sections, the issueof joint monopoly behaviour was one of the two main reasons for the introduction of  1 As to Bowley himself, he was probably more inclined to highlight the indeterminacy of the result.This at least is what can be argued from the second part of the previous quotation in the text, where heseemed to claim that the dependence of the solution upon the conjectures entailed that the systemwould ‘oscillate’, unless an explicit collusive agreement was reached. Hence collusion represented forBowley the way out from the indeterminacy caused by the conjectural term. 2 Cf. Tirole (1988, pp. 244-5); Varian (1992, p. 303). See, however, below, fn. 23[?]. The escape from conjectural variations 5
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